(a) Give the definitions of relation and equivalence relation on a set $S$.

(b) Let $\Sigma$ be the set of ordered pairs $(A, f)$ where $A$ is a non-empty subset of $\mathbb{R}$ and $f: A \rightarrow \mathbb{R}$. Let $\mathcal{R}$ be the relation on $\Sigma$ defined by requiring $(A, f) \mathcal{R}(B, g)$ if the following two conditions hold:

(i) $(A \backslash B) \cup(B \backslash A)$ is finite and

(ii) there is a finite set $F \subset A \cap B$ such that $f(x)=g(x)$ for all $x \in A \cap B \backslash F$.

Show that $\mathcal{R}$ is an equivalence relation on $\Sigma$.